Contemporary PET scanners for clinical use have spatial-resolution of 4 - 5 mm, caused by fundamental factors in medical imaging: detector sizes, free path of positrons, and non-colinearity uncertainty of annihilation photon-pairs. The drawback in resolution significantly restrained the sensitivity of PET in imaging small lesions, which could be either early-stage cancers or small metastasis. In this study, the principle for a novel scanning mode to acquire high spatial-resolution images is proposed for clinical PET scanners. The concept of equivalent position was first proposed as different angular orientations of the scanner ring, at which comparable images could be achieved. Due to this concept, a typical static PET scan can be separated into m ( m ≥ 2) equivalent sub-scans at different equivalent positions, when the scanner ring is systematically adjusted to m equivalent-positions of equal distance within one detector size. In this case each detector is virtually divided into m equal sub-detectors, without physical minimizing the detector size, and imaging contributions from every 1/ m part of the detector can be determined by an analytically matrix, since there are m variables and m sub-scans. This novel concept is quite feasible to contemporary design because the high spatial resolution working modes ( m ≥ 2) only demand the scanner to be slightly adjustable to other angular orientations. Adding high spatial resolutions modes to the scanner only has trifling influence on contrast resolutions as all imaging events at each sub-scan are independent. The time for performing a high-resolution scan could be comparable to a typical PET scan, as long as the Poisson noises are insignificant to low-uptake voxels. As a result, for a typical scanner design e.g. 80 cm in diameter with <sup>18</sup> F as tracers, the spatial resolution of double sub-scans ( m = 2) is 2.56 mm, and 2.19 mm for triple sub-scans ( m = 3), which are significant improvements. The novelty of high spatial resolution design is compatible to digital PET or any other technological evolutions.
Clinical positron emission tomography (PET) imaging has become indispensable in cardiology and cancer diagnosis since 1990s [
Spatial resolution of PET (denoted by R) is limited by multiple components, either solvable or fundamental [
The range of positron RR, (FWHM of range profile) is the mean distance between the radionuclide (where the β+ decay occurred) and the position where the positronium annihilated, thus RR depends on tissue densities and radionuclide types i.e. the energy of positrons. For 18F, RR = 0.54 mm in water for endpoint energy of 0.64 MeV [
In contemporary PET design, scintillating signals from detectors were collected and amplified by photomultiplier tubes (PMTs) into analog signals, rather than directly converted into digital signals. For discrete detectors of size d, the detector resolution is half of the detector size d/2.
Either RR or the detector size is more lucid than RNC, since RNC was not discovered until people learned that R actually became worse if one enlarged scanner diameter to achieve better resolution. RNC accounts for the non-zero net-momentum of positronium in the center-of-mass frame of reference. In another words, a positronium is in thermal motion like other molecules at body temperature, causing uncertainty in exit directions for annihilation photons. Obviously, cooling the patient would help since that reduced thermal motions for all particles in body; however, this would also tremendously alter the physiological processes (e.g. glucose metabolism) to be imaged in PET [
RNC = 0.0022 × D (1)
Although RNC is proportional to scanner diameter, shrinking scanner size for better RNC is not an option due to clinical requirement. The only exception might be the micro-PET scanners for small animal studies, which typically contains a compact scanner of 10 - 15 cm in diameter, and this leads to R ≈ 1 mm [
The overall spatial resolution can be expressed as:
R = R R 2 + R N C 2 + ( d / 2 ) 2 (2)
Since neither RR nor RNC could be possibly optimized in design, more efforts were put into detector resolution to improve whole spatial resolution. The choice of detector size is also a clinical consideration, because using smaller detectors causes low collecting efficiency, hence requiring more injection radionuclides/dose to the patients. In recent development of digital-PET, the analog PMTs in the traditional design were replaced by solid-state avalanche photodiodes (APD) coupled by digital acquisition system [
R = R R 2 + R N C 2 + R D E T 2 (2’)
As reported, RDET < d/2, the technological evolution of digital PET would certainly improve R, although more convincing results are expected in the future.
This feasibility study also attempted to improve detector response for better spatial resolution, as a parallel approach to digital-PET. Other than directly digitizing the scintillating signals, this study focused on high resolution working modes for clinical PET scanner, without significantly altering prevalent clinical PET design. Implementing high resolution working modes is compatible to any parallel technological evolution such as digital-PET, and could be potentially meaningful to any other emission-based tomography.
Suppose 2 clinical PET-scanners of the same type were manufactured, with each consisting of N identical detectors of size ρ as defined below, but the orientations of the detectors are slightly different in those 2 scanners. As shown in
Equivalent-positions are defined as positions of different orientation for the detector ring that can generate equivalent images (
For a typical PET scan using 18FDG tracer, the theoretical limit of spatial resolution would be 1.8 mm for an 80 cm-diameter scanner, if the detector size were minimal (RDET = 0). In contemporary design, the selection of detector size is of a
major concern on collecting efficiency, which also depends on engineering and electronic limits, e.g. collimator, deadtime, energy resolution etc. However, for any successful emission-based tomography, there always exists an optimal detector size, as a result of compromising collecting efficiency and spatial resolution. With any advancement of technological evolution such as digital-PET, the optimal detector size might also change. The mathematical relationship between optimal detector size and spatial resolution will be discussed elsewhere.
Since all PET scanners are cylindrical, it would be convenient to use angular size rather than physical size of detector. The angular size φ of each detector in a ring is: φ = |θ2 − θ1| = |θ3 −θ2|= … = |θN − θN−1| = 2π/N, and the optimal size of detector can be expressed in term of angular size: ρ = (D/2) φ, in which D is the diameter of the scanner.
For a PET scanner ring of N detectors, a PET image (denoted by F) is virtually the superposition of sub-images contributed by N individual detectors. However, remember that any event in PET imaging is a coincidence of 2 detectors, thus counting the events at each detector actually doubling the counts in statistics. Nevertheless, for convenience a PET image F can still be represented by superposition of N sub-images contributed by N individual detectors, but mathematically counted twice:
F = 1 2 ∑ i = 1 N f i (3)
The 1/2 at Equation (3) means the events were counted twice after checking all N detectors. Assume an adjustable PET scanner takes 2 images (at identical imaging conditions) FA and FB at 2 equivalent-positions ΘA and ΘB, respectively. Re-write Equation (3) for each equivalent-position respectively, sum up those 2 equations and then divide by 2 for each side. Given that F is the same for 2 images (FA = FB = F), we have:
F = 1 4 ∑ i = 1 N ( f i ( A ) + f i ( B ) ) (4)
In which f i ( A ) and f i ( B ) stand for the sub-images of the ith detector at ΘA and ΘB respectively.
Consider the simplest situation, in which the orientation difference between those 2 equivalent-positions is exactly half of the angular size of detectors φ, i.e. ΘB = ΘA + ε = ΘA + φ/2 (
f i ( A ) = f i , 1 + f i , 2 f i ( B ) = f i − 1 , 2 + f i , 1 (5)
Note f N + 1 , 2 = f 1 , 2 Equation (4) becomes:
F = 1 4 ∑ i = 1 N ( f i ( A ) + f i ( B ) ) = 1 2 ∑ i = 1 N ∑ k = 1 2 f i , k = 1 2 ∑ i = 1 N ( f i , 1 + f i , 2 ) (4’)
In another word, image F can be represented by 2N sub-images from 2N detectors (“sub-detector”), rather than N sub-images from N detectors. Please note that in Equation (4’) both fi,1 and fi,2 still use the same position i.e. θi of detector i, even though it is already known that fi,1 is from the lower half and fi,2 is from the upper half of the detector (
F ′ = 1 2 ∑ i = 1 N ( f ′ i , 1 + f ′ i , 2 ) = 1 2 ∑ j = 1 2 N f ′ j (6)
Compare Equation (3) and Equation (6), it is obvious to tell that after taking images at 2 equivalent-positions, the original PET image F which was contributed by N detectors of optimal size ρ, can be improved to F’ of better quality, contributed by 2N detectors of size ρ/2. The improvement in spatial resolution from single to double equivalent-position is shown in
Solving the matrix denoted by Equation (5), the sub-images at each sub-detector can be acquired analytically. A modeling algorithm on image manipulation is needed to fulfill this purpose at image/sub-image level, though the detail of modeling algorithms is obviously beyond the scope of this context. Replacing RDET by RDET/2 in Equation (2’), a clinical PET-scanner using 18F as tracers may improve the spatial resolution from 4 mm to 2.6 mm (
The method mentioned above is virtually for pixel-based or 2-dimensional imaging. To generate a 3-dimensional or voxel-based image of better spatial resolution, imaging at 2 longitudinal equivalent-positions is needed, such that the longitudinal spatial resolution is also improved. This could be easily achieved by moving the imaging table for a distance of half detector size, with the same mechanism denoted by Equations (3) to (6).
The more general cases are imaging at multiple equivalent positions, as long as the PET scanner is adjustable to those equivalent-positions. Since all detectors are instrumented symmetrically in the scanner ring, the maximal angular displacement to achieve all possible equivalent-positions is virtually φ, i.e. the size of given detectors. To perform imaging at m equivalent-positions, the scanner needs to adjust its orientation m − 1 times, with equal angular distances for each adjustment. If m = 1, the scanner is working at normal mode (at high spatial resolution modes m > 1) and hence there is no need to adjust its orientation.
Similar to imaging at double equivalent positions (m = 2), which was discussed at Equation (5) and Equation (6), for imaging at triple equivalent positions (m = 3), Equation (5) can be expanded to:
f i ( A ) = f i , 1 + f i , 2 + f i , 3 f i ( B ) = f i − 1 , 3 + f i , 1 + f i , 2 f i ( C ) = f i − 1 , 2 + f i − 1 , 3 + f i , 1 (5’)
No. of m (Equivalent positions) | R (mm) (R0 = 4.0) | R (mm) (R0 = 4.5) | R (mm) (R0 = 5.0) |
---|---|---|---|
1 | 4.00 | 4.50 | 5.00 |
2 | 2.56 | 2.76 | 2.97 |
3 | 2.19 | 2.29 | 2.41 |
4 | 2.04 | 2.11 | 2.18 |
5 | 1.97 | 2.02 | 2.06 |
For imaging at m equivalent-positions, each given detector virtually separates its contributions into m sub-detectors of known positions determined by those equivalent-positions. In this case m × N variables need to be solved if there are N detectors, however, these are always analytical because images were taken at m equivalent-positions and hence there have been m × N readouts. In a more general form of imaging at m equivalent-positions, Equation (4) can be re-written as:
F = 1 2 m ∑ i = 1 N ∑ k = 1 m f i ( k ) (7)
In Equation (7) f i ( k ) is the sub-image of the ith detector (i = 1, 2, …, N) imaging at the kth (k = 1, 2, …, m) equivalent-position, and similar to Equation (5) f i ( k ) can be represented by m terms:
( f i ( 1 ) f i ( 2 ) … f i ( m − 1 ) f i ( m ) ) = ( f i , 1 + f i , 2 + f i , 3 … + f i , m − 1 + f i , m f i − 1 , m + f i , 1 + f i , 2 … + f i , m − 2 + f i , m − 1 … f i − 1 , 3 + f i − 1 , 4 + … + f i , 1 + f i , 2 f i − 1 , 2 + f i − 1 , 3 + … + f i − 1 , m + f i , 1 ) (8)
Plug in Equation (8) into Equation (7), and re-label the items, i.e. f i , m → f ′ m , j , f i , m − 1 → f ′ m , j − 1 , …etc., we have:
F ′ = 1 2 ∑ j = 1 m N f ′ j (9)
Similar to Equation (6) for imaging at double equivalent positions, Equation (9) indicates that a PET scan F can be represented by m × N sub-images from m × N pseudo sub-detectors. Re-write Equation (2’), the overall spatial resolution for imaging at m equivalent-positions is:
R = R R 2 + R N C 2 + ( R D E T / m ) 2 (10)
Equation (10) shrinks to Equation (2’) if m = 1, which is for the normal working mode. The theoretical best spatial resolution (R = 1.8 mm for 18F) occurs when m → ∞ such that R D E T / m → 0 . However, imaging at many equivalent-positions cannot improve the spatial resolution too much.
Since the spatial resolution could be improved this way, it would be possible to design relatively larger scanners only by slightly reducing the spatial resolution. Larger scanners have the advantage of scanning patients of larger in size.
The pre-scan procedure for taking high-resolution scans is virtually identical to that for typical PET scans. Suppose a high-resolution mode (e.g. m = 2) is chosen, time slots for 2 sub-scans are assigned to 2 equivalent-positions by the system with consideration the trace decays, and the sub-scans at each equivalent-position are taken sequentially. The equivalent-positions should be evenly distributed within the angular size φ to ensure the precision of sub-detector size.
Since the PET scanner is cylindrically symmetric, equivalent positions at longitudinal direction are independent to equivalent positions at angular direction. With this concern, the total sub-scans at m equivalent-positions at both angular
and longitudinal directions would be m2 times (e.g. m2 = 4 if m = 2 is chosen) as normal working mode.
Unlike CT units, traditional PET scanners were stationary and not adjustable to different orientations. Working at high resolution modes also requests the scanner to be stationary while taking the high resolution PET scans. However, it also requires the scanner to be adjustable to different angular equivalent positions. This would slightly alter the traditional PET scanner design by adding the rotational capability for a maximal distance of 1 detector size.
The essential premise for applying multi-equivalent-position imaging is that positron emissions are stable, which typically occurred after some uptake time e.g. 45 min. This premise actually has been taken for all static PET imaging. Before taking a static PET scan, the patient is normally sent to the waiting room for 45 - 60 minutes after injecting the tracer e.g. 18FDG, and this waiting period ensures sufficient uptakes in the targets. Imaging at 45 minutes or 60 minutes after injection are normally assumed equivalent, since the uptake process has already completed thus there is no need to consider time-dependent uptakes.
Dynamic imaging sometimes is needed, mostly for research purposes. For instance, to discriminate necrosis from hypoxic volumes by the timing responses to the tracers, a dynamic PET scan rather than static PET is normally taken, from the injection moment up to at least 1 hour after injection. In another words, the goal of a dynamic scan is to image the whole uptake process, in which the emission rate for the targets is time-dependent, i.e. changing sharply before it eventually stabilizes. Imaging at high resolution working modes is virtually a special type of static PET imaging, and hence has drawbacks under this scenario. If all sub-images become time dependent, i.e. f i ( A ) → f i ( A ) ( t 1 ) and f i ( B ) → f i ( B ) ( t 2 ) , the contributions from sub-detectors are no longer analytical. The difficulties of applying high spatial resolution working modes to dynamic imaging are still unclear, although perspective studies are expected for the needs of establishing successful analytical methods.
A clinical PET scan typically takes 15 - 20 minutes. Imaging at high spatial resolution modes might demand extended scan time, because in each sub-image at equivalent positions the signal to noise ratio might drop since the number of events drop to 1/m whereas the noise level remains unchanged.
Another major concern would be the Poisson noise for some low-uptake voxels in a given sub-scan at an equivalent-position, which could be avoid by (a) increasing the imaging time at each equivalent-position; or (b) increase the injection amount of tracers to the patient. Option (a) is feasible as long as patient immobilization during a moderately longer scan is properly handled. In addition, organ motions with time should also be considered during longer time. A suitable scan time should be a compromise of both minimizing the Poisson noise and avoiding unnecessary motion artifacts. Option (b) is easier however generally not recommended since it increases the patient exposure, unless for some critical situations when benefit to risk analysis is carefully performed.
For some indispensable clinical demands, such as the minimal scanner size to accommodate most patients, and the minimal detector size to keep a reasonable collecting efficiency to avoid too much patient dose, prevalent clinical PET scanner seems to reach its limit on spatial resolution around 4 - 5 mm. Since neither the free-path uncertainty nor the non-colinearity uncertainty could be possibly optimized in clinical PET scanner, the only breakthrough seems to be the efforts on detector resolution. Other than technological evolution such as digital PET, imaging at high spatial resolution working modes is another parallel approach which provides great potential on PET advancement.
In this study, the concept of equivalent position of imaging was proposed for the first time. Imaging at those positions are deemed equivalent. Implementing this concept in imaging and reconstruction, a static PET scan could be considered as superposition of equivalent scans at different angular orientations without varying imaging quality and radiobiological information. It could be concluded in theory that high spatial resolution working modes for PET scanners are feasible at imaging manipulation level, after imaging at more than one equivalent imaging position and analyzing the contribution of sub-detectors with more precise positions. The improvement in spatial resolution is significant at m = 2 (or double-imaging) mode although imaging at multiple equivalent positions (m > 2) would enhance the resolution more. For typical detector size of 7 - 10 mm, the spatial resolution at m = 2 could become 2.6 - 3.0 mm from 4.0 to 5.0 mm. For this reason, imaging at double equivalent positions is comparatively recommended.
Under high spatial resolution modes, a static PET scan would be capable of imaging much smaller lesions, which is significant on imaging tumors at early stages, and hence greatly improve the opportunities on cancer controls.
The author would gratefully acknowledge Dr. Mutian Zhang for his helpful comments on the development of digital PET scanner.
There is no source of funding for this study.
There were no human subjects or animal subjects involved in this study.
There is no conflict of interests.
Wang, K. (2018) Feasibility of High Spatial Resolution Working Modes for Clinical PET Scanner. International Journal of Medical Physics, Clinical Engineering and Radiation Oncology, 7, 539-552. https://doi.org/10.4236/ijmpcero.2018.74045